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2008•Birkhäuser Boston eBooksRequires access

Invariants from the Seifert Matrix

Kunio Murasugi

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Abstract

In order to find a knot (or link) invariant from a Seifert matrix, we need to look for something that will not change under the operations Λ1 and \(\Lambda^{\pm 1}_2\), defined in Theorem 5.4.1. We will see in this chapter that the Alexander polynomial is such an invariant. The Alexander polynomial is not the only important invariant that we can extricate from the Seifert matrix, the signature of a link can also be defined from it. In addition to defining these two invariants we shall, in this chapter, prove some of their basic characteristics. Nota bene, throughout this chapter we shall assume all the knots and links are oriented.

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What this paper is about

In order to find a knot (or link) invariant from a Seifert matrix, we need to look for something that will not change under the operations Λ1 and \(\Lambda^{\pm 1}_2\), defined in Theorem 5.4.1. We will see in this chapter that the Alexander polynomial is such an invariant. The Alexander polynomial is not the only important invariant that we can extricate from the Seifert matrix, the signature of a link can also be defined from it. In addition to defining these two invariants we shall, in this chapter, prove some of their basic characteristics. Nota bene, throughout this chapter we shall assume all the knots and links are oriented.

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Available abstract

In order to find a knot (or link) invariant from a Seifert matrix, we need to look for something that will not change under the operations Λ1 and \(\Lambda^{\pm 1}_2\), defined in Theorem 5.4.1. We will see in this chapter that the Alexander polynomial is such an invariant. The Alexander polynomial is not the only important invariant that we can extricate from the Seifert matrix, the signature of a link can also be defined from it. In addition to defining these two invariants we shall, in this chapter, prove some of their basic characteristics. Nota bene, throughout this chapter we shall assume all the knots and links are oriented.

Key concepts: Alexander polynomial, Seifert surface, Invariant (physics), Knot (papermaking), Mathematics, Signature (topology), Pure mathematics, Matrix (chemical analysis)

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